Steady-State Response of a Liquid-Level System: 23-Mechatronics-A1 Systems Dynamics and Controls, Undated paper, Q3

Question 3: Steady-State Response of a Liquid-Level System

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

Given. A unity-feedback block diagram with forward path $K_p$ in series with a first-order plant $R/(RCs+1)$; the reference input is a step of amplitude $H_R$, i.e. $h_r(t)=H_Ru(t)$.

Find. The steady-state value $h_{css}(t)=\lim_{t\to\infty}h_c(t)$.

Figure — unity-feedback liquid-level control loop: gain $K_p$ cascaded with the tank's first-order plant $R/(RCs+1)$.

Approach. Form the closed-loop transfer function $H_c(s)/H_r(s)$, multiply by the step input's Laplace transform, and apply the Final Value Theorem.

Closed-loop transfer function (unity feedback). $$\frac{H_c(s)}{H_r(s)}=\frac{K_pR/(RCs+1)}{1+K_pR/(RCs+1)}=\frac{K_pR}{RCs+1+K_pR}$$
Apply the step input $H_r(s)=H_R/s$. $$H_c(s)=\frac{K_pR\,H_R}{s\,(RCs+1+K_pR)}$$
Final Value Theorem (valid since the closed-loop pole $s=-(1+K_pR)/(RC)$ is in the left half-plane for any physically-realizable positive $R,C,K_p$): $$h_{css}=\lim_{s\to 0}sH_c(s)=\boxed{h_{css}=\dfrac{K_pR}{1+K_pR}\,H_R}$$

Quantity	Expression
Closed-loop pole	$s=-(1+K_pR)/(RC)$
Steady-state response	$h_{css}=K_pR\,H_R/(1+K_pR)$

Concept & Theory

Topic: Steady-state response of a unity-feedback first-order (Type-0) system to a step input.

Key relations: Final Value Theorem $\lim_{t\to\infty}f(t)=\lim_{s\to 0}sF(s)$ (valid only when $sF(s)$ has all poles in the open left half-plane); unity-feedback closed-loop transfer function $G/(1+G)$.

Why this works: only the DC ($s=0$) value of the closed-loop transfer function sets the eventual level, because the $(RCs+1)$ dynamics decay as $e^{-t/\tau}$; since the loop has no free integrator (Type-0), it can never track a step with zero error, only reduce the error as the proportional gain $K_p$ grows large.

Common pitfall: applying the Final Value Theorem to a signal whose transform has a right-half-plane or imaginary-axis pole (it silently returns a meaningless finite number); forgetting the loop is unity-feedback and mis-forming the closed-loop transfer function.

Source: Ogata, Modern Control Engineering, §5-3 (steady-state errors); Nise, Control Systems Engineering, §7.2.

Question 3 of 6: Steady-State Response of a Liquid-Level System

Question 3: Steady-State Response of a Liquid-Level System